When you are doing mathematical integration over a function which is uniquely defined at every point, the intervals can be infinitely small with no quantum effects. It helps greatly if the function is continuous, with continuous derivatives.

If you are trying to integrate a function which has no defined derivative, and the values are not unique (like some fractals), then there is an element of uncertainty involved, and probabilities may be one way to resolve this.

If you are trying to calculate the values of physical systems, which *are* subject to quantum effects, then you must include these quantum effects in your evaluation of the integral. You can use probabilities to produce an "average" answer, or more rigorous methods to produce a distribution of answers.